Discrete and Ultradiscrete Periodic Phase Soliton Equations

We propose new type of discrete and ultradiscrete soliton equations, which admit extended soliton solution called periodic phase soliton solution. The discrete equation is derived from the discrete DKP equation and the ultradiscrete one is obtained by applying the ultradiscrete limit. The soliton solutions have internal freedom and change their shape periodically during propagation. In particular, the ultradiscrete solution reduces into the solution to the ultradiscrete hungry Lotka-Volterra equation in a special case.